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A386935 Integers with the same arithmetic mean for divisors and anti-divisors.

%I A386935 #27 Aug 12 2025 12:03:45
%S A386935 3,15,135,376,6956,1913646,1838558856
%N A386935 Integers with the same arithmetic mean for divisors and anti-divisors.
%C A386935 For the listed numbers the arithmetic means are 2, 6, 30, 90, 1064, 97128, 143824680, ...
%C A386935 a(8) > 10^10, if it exists. - _Amiram Eldar_, Aug 12 2025
%e A386935 Divisors of 135 are 8: 1, 3, 5, 9, 15, 27, 45, 135. Their sum is 240 and 240/8 = 30.
%e A386935 Anti-divisors of 135 are 7: 2, 6, 10, 18, 30, 54, 90. Their sum is 210 and 210/7 = 30.
%p A386935 with(numtheory): P:=proc(q) local a, b, k, n, v; v:=[];
%p A386935 for n from 3 to q do k:=2; a:=0; b:=0;
%p A386935 for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then b:=b+1; a:=a+k; fi; od;
%p A386935 if sigma(n)/tau(n)=a/b then v:=[op(v), n]; fi; od; op(v); end: P(10^4);
%o A386935 (Python)
%o A386935 from itertools import count, islice
%o A386935 from sympy.ntheory.factor_ import divisor_sigma, antidivisors
%o A386935 def A386935_gen(startvalue=3): # generator of terms >= startvalue
%o A386935     for k in count(max(startvalue,3)):
%o A386935         if divisor_sigma(k)*len(d:=antidivisors(k))==divisor_sigma(k,0)*sum(d):
%o A386935             yield k
%o A386935 A386935_list = list(islice(A386935_gen(),5)) # _Chai Wah Wu_, Aug 12 2025
%Y A386935 Cf. A000203/A000005, A066417/A066272.
%Y A386935 Cf. A003601, A192284.
%K A386935 nonn,hard,more
%O A386935 1,1
%A A386935 _Paolo P. Lava_, Aug 09 2025
%E A386935 a(6) from _Michel Marcus_, Aug 09 2025
%E A386935 a(7) from _Amiram Eldar_, Aug 10 2025